7 In crazy golf, a golf ball is hit so that it starts to move in a vertical circle on the inside of a smooth cylinder.

Model the golf ball as a particle, $P$, of mass $m$. The circular path of the golf ball has radius $a$ and centre $O$. At time $t$, the angle between $O P$ and the horizontal is $\theta$, as shown in the diagram.

The golf ball has speed $u$ at the lowest point of its circular path.\\
\includegraphics[max width=\textwidth, alt={}, center]{9cfa110c-ee11-447a-b21a-3f436432e27d-6_739_742_719_641}
\begin{enumerate}[label=(\alph*)]
\item Show that, while the golf ball is in contact with the cylinder, the reaction of the cylinder on the golf ball is

$$\frac { m u ^ { 2 } } { a } - 3 m g \sin \theta - 2 m g$$
\item Given that $u = \sqrt { 3 a g }$, the golf ball will not complete a vertical circle inside the cylinder. Find the angle which $O P$ makes with the horizontal when the golf ball leaves the surface of the cylinder.\\
(4 marks)
\end{enumerate}

\hfill \mbox{\textit{AQA M2 2009 Q7 [10]}}